√335 at a glance
- Exact value
- √335
- Decimal (10 places)
- 18.3030052177
- Rounded
- 18.3 · 18.30 · 18.303
- Perfect square?
- No — between 18² and 19²
- Rational?
- Irrational
- Both square roots
- ±18.303005
- Prime factorization
- 5 × 67
- Cube root
- 6.945150
How to simplify √335
The prime factorization of 335 is 5 × 67. Every prime appears only once, so there is no pair to bring outside the radical — √335 is already in simplest form (a "square-free" radicand).
A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 335, 5 and 67 appear an odd number of times, so √335 is irrational and 18.3030052177 is a rounded value.
Where √335 sits between perfect squares
324 = 18² and 361 = 19² are the nearest perfect squares, so √335 lies between 18 and 19. 335 is 11 above 324 and 26 below 361, so the root is closer to 18.
- Straight line between 324 and 361: 18.2973 (0.03% low)
- Tangent from 18, i.e. 18 + 11 ÷ 36: 18.3056 (0.01% high)
- Tangent from 19, i.e. 19 − 26 ÷ 38: 18.3158 (0.07% high)
For √335 the tangent at 18 wins, missing by only 0.0026. Tangent estimates shine when the number sits close to a perfect square — here 335 is just 11 above 324.
Finding √335 with the Babylonian method
If a guess is too big, 335 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√335) in one step.
Start from the nearest whole number, 18 (18² = 324):
| Step | Guess x | 335 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 18.0000000000 | 18.6111111111 | 18.3055555556 | 2 |
| 2 | 18.3055555556 | 18.3004552352 | 18.3030053954 | 6 |
| 3 | 18.3030053954 | 18.3030050401 | 18.3030052177 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √335 = 18.3030052177 to every decimal shown.
√335 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √335 the pattern is [18; 3, 3, 3, 36] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √335 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 18/1 | 18.0000000000 | 3.0 × 10⁻¹ |
| 55/3 | 18.3333333333 | 3.0 × 10⁻² |
| 183/10 | 18.3000000000 | 3.0 × 10⁻³ |
| 604/33 | 18.3030303030 | 2.5 × 10⁻⁵ |
| 21,927/1,198 | 18.3030050083 | 2.1 × 10⁻⁷ |
| 66,385/3,627 | 18.3030052385 | 2.1 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 335y² = 1. Its smallest solution in positive whole numbers is x = 604, y = 33.
√335 in geometry and everyday measurements
- A square patio or deck of 335 square feet is about 18.3 ft (18 ft 4 in) on each side, so edging all the way around takes 4 × √335 ≈ 73.2 ft.
- 335 is not a sum of two whole-number squares — the prime factor 67 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √335 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √335 as its space diagonal.
Square roots near √335 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √332 | 2√83 | 18.2209 | No |
| √333 | 3√37 | 18.2483 | No |
| √334 | √334 | 18.2757 | No |
| √335 | √335 | 18.3030 | No |
| √336 | 4√21 | 18.3303 | No |
| √337 | √337 | 18.3576 | No |
| √338 | 13√2 | 18.3848 | No |
- The cube root of 335 is about 6.945150.
- Squaring undoes the root: (√335)² = 335, while 335² = 112,225 — the number whose square root is 335.
Frequently asked questions
What is the square root of 335?
The square root of 335 is √335, about 18.3030052177. The negative root, −18.303005, also squares to 335.
Is the square root of 335 rational or irrational?
Irrational. 335 is not a perfect square — it falls between 324 and 361 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √335 be simplified?
No. 335 = 5 × 67 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √335 rounded to two decimal places?
√335 ≈ 18.30 to two decimal places (18.3 to one, 18.303 to three). Check: 18.30² = 334.89, close to 335.